Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity

Contents
Preface
Acknowledgements
1. Introduction
	1.1 A Vector Space Approach to Euclidean Geometry and A Gyrovector Space Approach to Hyperbolic 
Geometry
	1.2 Gyrolanguage
	1.3 Analytic Hyperbolic Geometry
	1.4 The Three Models
	1.5 Applications in Quantum and Special Relativity Theory
2. Gyrogroups
	2.1 Definitions
	2.2 First Gyrogroup Theorems
	2.3 The Associative Gyropolygonal Gyroaddition
	2.4 Two Basic Gyrogroup Equations and Cancellation 
Laws
	2.5 Commuting Automorphisms with Gyroautomorphisms
	2.6 The Gyrosemidirect Product Group
	2.7 Basic Gyration Properties
3. Gyrocommutative Gyrogroups
	3.1 Gyrocommutative Gyrogroups
	3.2 Nested Gyroautomorphism Identities
	3.3 Two-Divisible Two-Torsion Free Gyrocommutative Gyrogroups
	3.4 From Möbius to Gyrogroups
	3.5 Higher Dimensional Möbius Gyrogroups
	3.6 Möbius Gyrations
	3.7 Three-Dimensional Möbius
 gyrations
	3.8 Einstein Gyrogroups
	3.9 Einstein Gyrations
	3.10 Einstein Coaddition
	3.11 PV Gyrogroups
	3.12 Points and Vectors in a Real Inner Product Space
	3.13 Exercises
4. Gyrogroup Extension
	4.1 Gyrogroup Extension
	4.2 The Gyroinner Product, the Gyronorm, and the Gyroboost
	4.3 Extended Automorphisms
	4.4 Gyrotransformation Groups
	4.5 Einstein Gyrotransformation Groups
	4.6 PV (Proper Velocity) Gyrotransformation Groups
	4.7 Galilei Transformation Groups
	4.8 From Gyroboosts to Boosts
	4.9 Lorentz Boost
	4.10 The (p :q)-Gyromidpoint
	4.11 The (p1 :p2 : . . . : pn)-Gyromidpoint
5. Gyrovectors and Cogyrovectors
	5.1 Equivalence Classes
	5.2 Gyrovectors
	5.3 Gyrovector Translation
	5.4 Gyrovector Translation Composition
	5.5 Points and Gyrovectors
	5.6 The Gyroparallelogram Addition Law
	5.7 Cogyrovectors
	5.8 Cogyrovector Translation
	5.9 Cogyrovector Translation Composition
	5.10 Points and Cogyrovectors
	5.11 Exercises
6. Gyrovector Spaces
	6.1 Definition and First Gyrovector Space Theorems
	6.2 Solving a System of Two Equations in a Gyrovector Space
	6.3 Gyrolines and Cogyrolines
	6.4 Gyrolines
	6.5 Gyromidpoints
	6.6 Gyrocovariance
	6.7 Gyroparallelograms
	6.8 Gyrogeodesics
	6.9 Cogyrolines
	6.10 Carrier Cogyrolines of Cogyrovectors
	6.11 Cogyromidpoints
	6.12 Cogyrogeodesics
	6.13 Various Gyrolines That Correspond to Cancellation Laws
	6.14 Möbius Gyrovector Spaces
	6.15 Möbius Cogyroline Parallelism
	6.16 Illustrating the Gyroline Gyration Transitive Law
	6.17 Turning Möbius Gyrometric into Poincaré Metric
	6.18 Einstein Gyrovector Spaces
	6.19 Turning Einstein Gyrometric into a Metric
	6.20 PV (Proper Velocity) Gyrovector Spaces
	6.21 Gyrovector Space Isomorphisms
	6.22 Converting the Einstein Half into the PV Half
	6.23 Gyrotriangle Gyromedians and Gyrocentroids
		6.23.1 In Einstein Gyrovector Spaces
		6.23.2 In Möbius Gyrovector Spaces
		6.23.3 In PV Gyrovector Spaces
	6.24 Exercises
7. Rudiments of Differential Geometry
	7.1 The Riemannian Line Element of Euclidean Metric
	7.2 The Gyroline and the Cogyroline Element
	7.3 The Gyroline Element of Möbius Gyrovector Spaces
	7.4 The Cogyroline Element of Möbius Gyrovector Spaces
	7.5 The Gyroline Element of Einstein Gyrovector Spaces
	7.6 The Cogyroline Element of Einstein Gyrovector Spaces
	7.7 The Gyroline Element of PV Gyrovector Spaces
	7.8 The Cogyroline Element of PV Gyrovector Spaces
	7.9 Table of Riemannian Line Elements
8. Gyrotrigonometry
	8.1 Vectors and Gyrovectors are Equivalence Classes
	8.2 Gyroangles
	8.3 Gyrovector Translation of Gyrorays
	8.4 Gyrorays Parallelism and Perpendicularity
	8.5 Gyrotrigonometry in Möbius Gyrovector Spaces
	8.6 Gyrotriangle Gyroangles and Side Gyrolengths
	8.7 The Gyroangular Defect of Right-Gyroangled Gyrotriangles
	8.8 Gyroangular Defect of the Gyrotriangle
	8.9 Gyroangular Defect of the Gyrotriangle — a Synthetic Proof
	8.10 The Gyrotriangle Side Gyrolengths in Terms of Its Gyroangles
	8.11 The Semi-Gyrocircle Gyrotriangle
	8.12 Gyrotriangular Gyration and Defect
	8.13 The Equilateral Gyrotriangle
	8.14 The Möbius Gyroparallelogram
	8.15 Gyrotriangle Defect in the Möbius Gyroparallelogram
	8.16 Gyroparallelograms Inscribed in a Gyroparallelogram
	8.17 Möbius Gyroparallelogram Addition Law
	8.18 The Gyrosquare
	8.19 Equidefect Gyrotriangles
	8.20 Parallel Transport
	8.21 Parallel Transport vs. Gyrovector Translation
	8.22 From Parallel Transports to Binary Operations
	8.23 Gyrocircle Gyrotrigonometry
	8.24 Cogyroangles
	8.25 The Cogyroangle in the Three Models
	8.26 Parallelism in Gyrovector Spaces
	8.27 Reflection, Gyroreflection, and Cogyroreflection
	8.28 Tessellation of the Poincaré Disc
	8.29 Bifurcation Approach to Non-Euclidean Geometry
	8.30 Exercises
9. Bloch Gyrovector of Quantum Information and Computation
	9.1 Bloch Vector and the Density Matrix for Mixed State Qubits
	9.2 Bloch Gyrovector
		9.2.1 Example 1
		9.2.2 Example 2
		9.2.3 Example 3
		9.2.4 Example 4
	9.3 Structure of the Qubit Density Matrix Space
	9.4 Trace Distance and Bures Fidelity
	9.5 Real Density Matrix for Mixed State Qubits
	9.6 Extending the Real Density Matrix
	9.7 Exercises
10. Special Theory of Relativity: The Analytic Hyperbolic 
Geometric Viewpoint
	10.1 Introduction
	10.2 Einstein Velocity Addition
	10.3 From Thomas Gyration to Thomas Precession
	10.4 The Relativistic Gyrovector Space
	10.5 Gyrogeodesics, Gyromidpoints and Gyrocentroids
	10.6 The Midpoint and the Gyromidpoint — Newtonian and Einsteinian Mechanical Interpretation
	10.7 Einstein Gyroparallelograms
	10.8 The Gyroparallelogram Addition Law Is Experimentally Significant
	10.9 Extending the Relativistic Gyroparallelogram Law
	10.10 The Parallelepiped
	10.11 The Pre-Gyroparallelepiped
	10.12 The Gyroparallelepiped
	10.13 The Relativistic Gyroparallelepiped Addition Law
11. Special Theory of Relativity: The Analytic Hyperbolic 
Geometric Viewpoint
	11.1 The Lorentz Transformation and Its Gyro-Algebra
	11.2 Galilei and Lorentz Transformation Links
	11.3 (t1: t2)-Gyromidpoints as CMM Velocities
	11.4 The Hyperbolic Theorems of Ceva and Menelaus
	11.5 Relativistic Two-Particle Systems
	11.6 The Covariant Relativistic CMM Frame Velocity
	11.7 The Relativistic Invariant Mass of an Isolated Particle System
	11.8 On the Relativistic Particle System Mass Theorem
	11.9 The Relativistic Law of Momentum for Particle Systems
	11.10 Relativistic CMM and the Kinetic Energy Theorem
	11.11 Additivity of Relativistic Energy and Momentum
	11.12 Newtonian and Relativistic Kinetic Energy
		11.12.1 The Newtonian Kinetic Energy
		11.12.2 The Relativistic Kinetic Energy
		11.12.3 An Unexpected Analogy That Classical and Relativistic Kinetic Energy Share
		11.12.4 Consequences of Classical Kinetic Energy Conservation During Elastic Collisions
		11.12.5 Consequences of Relativistic Kinetic Energy Conservation During Elastic Collisions
		11.12.6 On the Analogies and a Seeming Disanalogy
	11.13 Barycentric Coordinates
	11.14 Einsteinian Gyrobarycentric Coordinates
	11.15 The Proper Velocity Lorentz Group
	11.16 Demystifying the Proper Velocity Lorentz Group
	11.17 The Standard Lorentz Transformation Revisited
	11.18 The Inhomogeneous Lorentz Transformation
	11.19 The Relativistic Center of Momentum and Mass
	11.20 Relativistic Center of Mass: Example 1
	11.21 Relativistic Center of Mass: Example 2
	11.22 Dark Matter and Dark Energy
	11.23 Exercises
12. Relativistic Gyrotrigonometry
	12.1 The Relativistic Gyrotriangle
	12.2 Law of Gyrocosines, SSS to AAA Conversion Law
	12.3 The AAA to SSS Conversion Law
	12.4 The Law of Gyrosines
	12.5 The Relativistic Equilateral Gyrotriangle
	12.6 The Relativistic Gyrosquare
	12.7 The Relativistic Gyrosquare with θ = π/3
	12.8 The ASA to SAS Conversion Law
	12.9 The Relativistic Gyrotriangle Defect
	12.10 The Right Gyrotriangle
	12.11 Einsteinian Gyrotrigonometry
	12.12 Gyrodistance Between a Point and a Gyroline
	12.13 Gyrotriangle Gyroaltitudes
	12.14 Einstein Gyrotriangle Gyroorthocenter
	12.15 Möbius Gyrotriangle Gyroorthocenter
	12.16 Relativistic Gyrotriangle Gyroarea
	12.17 Gyrotriangle Gyroarea Addition
	12.18 Gyrosquare Gyroarea
	12.19 The Gyrotriangle Constant Principle
	12.20 Ceva and Menelaus, Revisited
	12.21 Saccheri Gyroquadrilaterals
	12.22 Lambert Gyroquadrilaterals
	12.23 Gyrotetrahedron Gyroaltitudes
	12.24 Exercises
13. Stellar and Particle Aberration
	13.1 Particle Aberration: The Classical Interpretation
	13.2 Particle Aberration: The Relativistic Interpretation
	13.3 Particle Aberration: The Geometric Interpretation
	13.4 Relativistic Stellar Aberration
	13.5 Exercises
14. Enriched Special Relativity Theory: Special Relativity of 
Signature (m,n)
	14.1 Introduction
	14.2 Galilei and Lorentz Boosts and Multi-boosts
	14.3 Pseudo-Euclidean Spaces and Lorentz Transformations of Any Signature
	14.4 Matrix Balls of Radius c
	14.5 Bi-gamma Factor
	14.6 V-Parametric Realization of Lorentz Transformations of Signature (m,n)
	14.7 Additive Decomposition of the Lorentz Bi-boost
	14.8 Application of the Galilei Bi-boost of Signature (1,3)
	14.9 Application of the Galilei Bi-boost of Any Signature
	14.10 Application of the Lorentz Bi-boost of Any Signature
	14.11 Lorentz Bi-boost Composition Law
	14.12 A Supporting Conjecture
	14.13 Epilogue
	14.14 Exercises
Notation and Special Symbols
Bibliography
Index